Topologies on quantum effects (Q962561)

The authors study various topologies on the effect algebra \({\mathcal E}(H)\) of effects on a Hilbert space \(H\) and on the orthomodular lattice \({\mathcal P}(H)\) of projections on a Hilbert space \(H\). Besides the norm topology and weak and strong operator topologies also topologies given by the partial order -- the order topology and the interval topology -- are studied. They conclude that all these topologies coincide if \(\dim H = 1\), that the interval topology is strictly weaker than the weak operator topology and the interval topology is strictly stronger than the strong operator topology if \(\dim H >1\), and that weak operator topology coincide with strong operator topology if and only if either \(H\) is finite-dimensional (in that case these topologies coincide also with the norm topology) or if \(P(H)\) is considered.